On Positivity of Fourier Transforms
2006 ◽
Vol 74
(1)
◽
pp. 133-138
◽
Keyword(s):
The Real
◽
This note concerns Fourier transforms on the real positive line. In particular, we seek conditions on a real function u(x) in x > 0, that ensure that its Fourier-cosine transform v(t) = u(x) cos xt dx is positive. We prove first that this is so for all t > 0, if u"(x) > 0 for all x > 0, that is, that everywhere-convex functions have everywhere-positive Fourier-cosine transforms. We then obtain a complex-plane criterion for some types of non-convex u(x). Finally we consider criteria on u(x) that imply positivity of v(t) for t > t0, for some t0 > 0.
Keyword(s):
2013 ◽
Vol 06
(01)
◽
pp. 1350005
◽
1988 ◽
Vol 1
(3)
◽
pp. 307-310
◽
Keyword(s):
Keyword(s):
1994 ◽
Vol 18
(6)
◽
pp. 343-350
◽
1977 ◽
Vol 18
(2)
◽
pp. 175-177
◽
2008 ◽
Vol 35
(10)
◽
pp. 1878-1881
◽